Exact and Inexact Graph Matching: Methodology and Applications

Graphs provide us with a powerful and flexible representation formalism which can be employed in various fields of intelligent information processing. The process of evaluating the similarity of graphs is referred to as graph matching. Two approaches to this task exist, viz. exact and inexact graph matching. The former approach aims at finding a strict correspondence between two graphs to be matched, while the latter is able to cope with errors and measures the difference of two graphs in a broader sense. The present chapter reviews some fundamental concepts of both paradigms and shows two recent applications of graph matching in the fields of information retrieval and pattern recognition.

[1]  Edwin R. Hancock,et al.  Computing approximate tree edit distance using relaxation labeling , 2003, Pattern Recognit. Lett..

[2]  Horst Bunke,et al.  A Graph-Theoretic Approach to Enterprise Network Dynamics (Progress in Computer Science and Applied Logic (PCS)) , 2006 .

[3]  Steven Gold,et al.  A Graduated Assignment Algorithm for Graph Matching , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  Javier Larrosa,et al.  Constraint satisfaction algorithms for graph pattern matching , 2002, Mathematical Structures in Computer Science.

[5]  Eam Khwang Teoh,et al.  Pattern recognition by graph matching using the Potts MFT neural networks , 1995, Pattern Recognit..

[6]  Marcello Pelillo,et al.  Replicator Equations, Maximal Cliques, and Graph Isomorphism , 1998, Neural Computation.

[7]  Shinji Umeyama,et al.  An Eigendecomposition Approach to Weighted Graph Matching Problems , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  A. John MINING GRAPH DATA , 2022 .

[9]  Benoit Huet,et al.  Shape recognition from large image libraries by inexact graph matching , 1999, Pattern Recognit. Lett..

[10]  Michael J. Fischer,et al.  The String-to-String Correction Problem , 1974, JACM.

[11]  Josef Kittler,et al.  Combining Evidence in Probabilistic Relaxation , 1989, Int. J. Pattern Recognit. Artif. Intell..

[12]  Horst Bunke,et al.  A decision tree approach to graph and subgraph isomorphism detection , 1999, Pattern Recognit..

[13]  Ah Chung Tsoi,et al.  The Graph Neural Network Model , 2009, IEEE Transactions on Neural Networks.

[14]  Horst Bunke,et al.  Automatic learning of cost functions for graph edit distance , 2007, Inf. Sci..

[15]  Edwin R. Hancock,et al.  Structural Graph Matching Using the EM Algorithm and Singular Value Decomposition , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  Kaspar Riesen,et al.  Non-linear Transformations of Vector Space Embedded Graphs , 2008, PRIS.

[17]  Hanan Samet,et al.  Properties of Embedding Methods for Similarity Searching in Metric Spaces , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  G. Levi A note on the derivation of maximal common subgraphs of two directed or undirected graphs , 1973 .

[19]  John E. Hopcroft,et al.  Linear time algorithm for isomorphism of planar graphs (Preliminary Report) , 1974, STOC '74.

[20]  Alfred O. Hero,et al.  A binary linear programming formulation of the graph edit distance , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[21]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[22]  R. Fisher THE STATISTICAL UTILIZATION OF MULTIPLE MEASUREMENTS , 1938 .

[23]  Kristina Schädler,et al.  Comparing Structures Using a Hopfield-Style Neural Network , 1999, Applied Intelligence.

[24]  J. Munkres ALGORITHMS FOR THE ASSIGNMENT AND TRANSIORTATION tROBLEMS* , 1957 .

[25]  Julian R. Ullmann,et al.  An Algorithm for Subgraph Isomorphism , 1976, J. ACM.

[26]  Thomas Gärtner,et al.  On Graph Kernels: Hardness Results and Efficient Alternatives , 2003, COLT.

[27]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[28]  Marco Gori,et al.  Exact and approximate graph matching using random walks , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[29]  Kaspar Riesen,et al.  On Lipschitz Embeddings of Graphs , 2008, KES.

[30]  Radford M. Neal Pattern Recognition and Machine Learning , 2007, Technometrics.

[31]  Kaspar Riesen,et al.  Generalized Graph Matching for Data Mining and Information Retrieval , 2008, ICDM.

[32]  C. Watkins Dynamic Alignment Kernels , 1999 .

[33]  Horst Bunke,et al.  Inexact graph matching for structural pattern recognition , 1983, Pattern Recognit. Lett..

[34]  Edwin R. Hancock,et al.  Levenshtein distance for graph spectral features , 2004, Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004..

[35]  Terry Caelli,et al.  Inexact Graph Matching Using Eigen-Subspace Projection Clustering , 2004, Int. J. Pattern Recognit. Artif. Intell..

[36]  Shigeo Abe DrEng Pattern Classification , 2001, Springer London.

[37]  Horst Bunke,et al.  Self-organizing maps for learning the edit costs in graph matching , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[38]  Alessio Micheli,et al.  Neural Network for Graphs: A Contextual Constructive Approach , 2009, IEEE Transactions on Neural Networks.

[39]  Kaspar Riesen,et al.  Reducing the dimensionality of dissimilarity space embedding graph kernels , 2009, Eng. Appl. Artif. Intell..

[40]  Martin A. Fischler,et al.  The Representation and Matching of Pictorial Structures , 1973, IEEE Transactions on Computers.

[41]  Horst Bunke,et al.  Optimal quadratic-time isomorphism of ordered graphs , 1999, Pattern Recognit..

[42]  Horst Bunke,et al.  Matching graphs with unique node labels , 2004, Pattern Analysis and Applications.

[43]  Zaïd Harchaoui,et al.  Image Classification with Segmentation Graph Kernels , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[44]  Edwin R. Hancock,et al.  Spectral embedding of graphs , 2003, Pattern Recognit..

[45]  A. Paone,et al.  Discrete Time Relaxation Based on Direct Quadrature Methods for Volterra Integral Equations , 1999, Computing.

[46]  Alessandro Sperduti,et al.  Supervised neural networks for the classification of structures , 1997, IEEE Trans. Neural Networks.

[47]  J. J. McGregor,et al.  Backtrack search algorithms and the maximal common subgraph problem , 1982, Softw. Pract. Exp..

[48]  Hans-Peter Kriegel,et al.  Protein function prediction via graph kernels , 2005, ISMB.

[49]  King-Sun Fu,et al.  A graph distance measure for image analysis , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[50]  William J. Christmas,et al.  Structural Matching in Computer Vision Using Probabilistic Relaxation , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[51]  Ali Shokoufandeh,et al.  Indexing hierarchical structures using graph spectra , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[52]  Edwin R. Hancock,et al.  Pattern Vectors from Algebraic Graph Theory , 2005, IEEE Trans. Pattern Anal. Mach. Intell..

[53]  Horst Bunke,et al.  Graph Edit Distance with Node Splitting and Merging, and Its Application to Diatom Idenfication , 2003, GbRPR.

[54]  David Haussler,et al.  Convolution kernels on discrete structures , 1999 .

[55]  Josef Kittler,et al.  Discrete relaxation , 1990, Pattern Recognit..

[56]  Mario Vento,et al.  Thirty Years Of Graph Matching In Pattern Recognition , 2004, Int. J. Pattern Recognit. Artif. Intell..

[57]  Tariq S. Durrani,et al.  A RKHS Interpolator-Based Graph Matching Algorithm , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[58]  Anil K. Jain,et al.  Feature Selection: Evaluation, Application, and Small Sample Performance , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[59]  Edwin R. Hancock,et al.  A Riemannian approach to graph embedding , 2007, Pattern Recognit..

[60]  Kaspar Riesen,et al.  Approximate graph edit distance computation by means of bipartite graph matching , 2009, Image Vis. Comput..

[61]  Kaspar Riesen,et al.  Graph Classification Based on Vector Space Embedding , 2009, Int. J. Pattern Recognit. Artif. Intell..

[62]  Alessandro Sperduti,et al.  A general framework for adaptive processing of data structures , 1998, IEEE Trans. Neural Networks.

[63]  Horst Bunke,et al.  Transforming Strings to Vector Spaces Using Prototype Selection , 2006, SSPR/SPR.

[64]  W. Wallis,et al.  A Graph-Theoretic Approach to Enterprise Network Dynamics , 2006 .

[65]  Horst Bunke,et al.  A graph distance metric based on the maximal common subgraph , 1998, Pattern Recognit. Lett..

[66]  Bernhard Schölkopf,et al.  Nonlinear Component Analysis as a Kernel Eigenvalue Problem , 1998, Neural Computation.

[67]  Edwin R. Hancock,et al.  Structural Matching by Discrete Relaxation , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[68]  Tatsuya Akutsu,et al.  Graph Kernels for Molecular Structure-Activity Relationship Analysis with Support Vector Machines , 2005, J. Chem. Inf. Model..

[69]  Horst Bunke,et al.  Bridging the Gap between Graph Edit Distance and Kernel Machines , 2007, Series in Machine Perception and Artificial Intelligence.

[70]  James C. Bezdek,et al.  Nearest prototype classifier designs: An experimental study , 2001, Int. J. Intell. Syst..

[71]  Antonio Robles-Kelly,et al.  String Edit Distance, Random Walks And Graph Matching , 2002, Int. J. Pattern Recognit. Artif. Intell..

[72]  Kaspar Riesen,et al.  Fast Suboptimal Algorithms for the Computation of Graph Edit Distance , 2006, SSPR/SPR.

[73]  Horst Bunke,et al.  A Comparison of Algorithms for Maximum Common Subgraph on Randomly Connected Graphs , 2002, SSPR/SPR.

[74]  King-Sun Fu,et al.  A distance measure between attributed relational graphs for pattern recognition , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[75]  Edwin R. Hancock,et al.  Bayesian graph edit distance , 1999, Proceedings 10th International Conference on Image Analysis and Processing.

[76]  Yuan Yao,et al.  Combining flat and structured representations for fingerprint classification with recursive neural networks and support vector machines , 2003, Pattern Recognit..

[77]  Horst Bunke,et al.  On a relation between graph edit distance and maximum common subgraph , 1997, Pattern Recognit. Lett..

[78]  Fritz Wysotzki,et al.  Automorphism Partitioning with Neural Networks , 2004, Neural Processing Letters.

[79]  Jignesh M. Patel,et al.  TALE: A Tool for Approximate Large Graph Matching , 2008, 2008 IEEE 24th International Conference on Data Engineering.

[80]  David G. Stork,et al.  Pattern Classification , 1973 .

[81]  Kaspar Riesen,et al.  Dissimilarity Based Vector Space Embedding of Graphs Using Prototype Reduction Schemes , 2009, MLDM.

[82]  Horst Bunke,et al.  A Quadratic Programming Approach to the Graph Edit Distance Problem , 2007, GbRPR.

[83]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2003, ICTAI.

[84]  Eam Khwang Teoh,et al.  Pattern recognition by homomorphic graph matching using Hopfield neural networks , 1995, Image Vis. Comput..

[85]  Miro Kraetzl,et al.  Graph distances using graph union , 2001, Pattern Recognit. Lett..

[86]  Abraham Kandel,et al.  On the Minimum Common Supergraph of Two Graphs , 2000, Computing.

[87]  Bernhard Schölkopf,et al.  Dynamic Alignment Kernels , 2000 .

[88]  Lawrence B. Holder,et al.  Mining Graph Data: Cook/Mining Graph Data , 2006 .

[89]  Mario Vento,et al.  A (sub)graph isomorphism algorithm for matching large graphs , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[90]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[91]  Rui Xu,et al.  Survey of clustering algorithms , 2005, IEEE Transactions on Neural Networks.

[92]  David G. Stork,et al.  Pattern classification, 2nd Edition , 2000 .

[93]  Jean-Philippe Vert,et al.  Graph-Driven Feature Extraction From Microarray Data Using Diffusion Kernels and Kernel CCA , 2002, NIPS.

[94]  Bernhard Schölkopf,et al.  Learning with kernels , 2001 .

[95]  Robert P. W. Duin,et al.  The Dissimilarity Representation for Pattern Recognition - Foundations and Applications , 2005, Series in Machine Perception and Artificial Intelligence.

[96]  Gabriel Valiente,et al.  A graph distance metric combining maximum common subgraph and minimum common supergraph , 2001, Pattern Recognit. Lett..

[97]  John D. Lafferty,et al.  Diffusion Kernels on Graphs and Other Discrete Input Spaces , 2002, ICML.

[98]  Christine Solnon,et al.  Reactive Tabu Search for Measuring Graph Similarity , 2005, GbRPR.

[99]  Abraham Kandel,et al.  Graph-Theoretic Techniques for Web Content Mining , 2005, Series in Machine Perception and Artificial Intelligence.

[100]  F TichyWalter The string-to-string correction problem with block moves , 1984 .

[101]  Andrew K. C. Wong,et al.  Entropy and Distance of Random Graphs with Application to Structural Pattern Recognition , 1985, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[102]  Alexander J. Smola,et al.  Kernels and Regularization on Graphs , 2003, COLT.

[103]  J. van Leeuwen,et al.  Graph Based Representations in Pattern Recognition , 2003, Lecture Notes in Computer Science.

[104]  Alexander J. Smola,et al.  Learning with kernels , 1998 .

[105]  Kaspar Riesen,et al.  Kernel k-Means Clustering Applied to Vector Space Embeddings of Graphs , 2008, ANNPR.

[106]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[107]  Eam Khwang Teoh,et al.  Self-organizing Hopfield network for attributed relational graph matching , 1995, Image Vis. Comput..

[108]  Thomas Gärtner,et al.  Kernels for structured data , 2008, Series in Machine Perception and Artificial Intelligence.